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On incidence coloring and star arboricity of graphs. (English) Zbl 0871.05022
Summary: We show that the concept of incidence coloring introduced by R. A. Brualdi and J. J. Q. Massey [(*) Discrete Math. 122, No. 1-3, 51-58 (1993; Zbl 0790.05026)] is a special case of directed star arboricity, introduced by I. Algor and N. Alon [(**) Discrete Math. 75, No. 1-3, 11-22 (1989; Zbl 0684.05033)]. A conjecture in (*) concerning asymptotics of the incidence coloring number is solved in the negative following an example in (**). We generalize a result in [N. Alon, C. McDiarmid, and B. Reed, Combinatorica 12, No. 4, 375-380 (1992; Zbl 0780.05043)] concerning the star arboricity of graphs to the directed case by a slight modification of their proof, to give the same asymptotic bound as in the undirected case. As a result, we get tight asymptotic bounds for the maximum incidence coloring number of a graph in terms of its degree.

MSC:
05C15 Coloring of graphs and hypergraphs
05C35 Extremal problems in graph theory
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References:
[1] Akiyama, J; Kano, M, Path factors of a graph, () · Zbl 0605.05034
[2] Algor, I; Alon, N, The star arboricity of graphs, Discrete math., 75, 11-22, (1989) · Zbl 0684.05033
[3] Alon, N; McDiarmid, C; Reed, B, Star arboricity, Combinatorica, 12, 375-380, (1992) · Zbl 0780.05043
[4] Brualdi, R.A; Massey, J.J.Q, Incidence and strong edge colorings of graphs, Discrete math., 122, 51-58, (1993) · Zbl 0790.05026
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